First-principles prediction of two-dimensional copper borides

This paper is a contribution to the joint Physical Review Applied and Physical Review Materials collection titled Two-Dimensional Materials and Devices.

Borides of group-IB and group-IIB metals normally do not exist in the bulk form due to the small electronegativity difference and large mismatch in their atomic sizes. Contrary to these established rules, several metastable two-dimensional structures of copper boride were predicted from ab initio evolutionary searches, identifying that two structures are metallic whereas another one is strikingly a nodal line semimetal. The reduced dimensionality and intralayer coupling are responsible for the emergence of copper borides. This makes copper borides an attractive material for potential electronics applications.

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Figure 1

(a) The formation energy of all 2D-${\mathrm{Cu}}_{1-x}{\mathrm{B}}_x$ structures (marked in cyan circles) from evolutionary search. The thermodynamically stable structures are laid on the convex hull. The detail of the 2D-${\mathrm{Cu}}_{1-x}{\mathrm{C}}_x$ system is indicated by red squares and dash line. (b) and (c) The reference structures of copper monolayer and ${\mathrm{g}}_{1/8}$-borophene [12].

Figure 2

(a)–(c) The structure of ${\mathrm{Cu}}_7{\mathrm{B}}_{15}$, ${\mathrm{CuB}}_3$, and ${\mathrm{CuB}}_6$. Nonequivalent atoms are drawn in orange (Cu1), red (Cu2), blue (B1), cyan (B2), and green (B3), corresponding to the atomic positions in Table 1. (d) Phonon dispersion curves of ${\mathrm{Cu}}_7{\mathrm{B}}_{15}$ with the high symmetry points of $\mathrm{\Gamma }$ (0 0 0), M (0 0.5 0), K ($-0.333$ 0.667 0). (e) Phonon dispersion curves of ${\mathrm{CuB}}_3$ with the high symmetry points of $\mathrm{\Gamma }$ (0 0 0), B (0 $-0.5$ 0), D (0.5 $-0.5$ 0), Z (0.5 0 0). (f) Phonon dispersion curves of ${\mathrm{CuB}}_6$ with the high symmetry points of $\mathrm{\Gamma }$ (0 0 0), X (0.5 0 0), S (0.5 0.5 0), Y (0 0.5 0).

Figure 3

(a) Band structure of ${\mathrm{CuB}}_3$. (b) DOS of ${\mathrm{CuB}}_3$. (c) Band structure of ${\mathrm{CuB}}_6$. (d) DOS of ${\mathrm{CuB}}_6$. (e) Band structure of the ${\mathrm{CuB}}_6$ with electron doping concentration of 1.76 $\times {10}^{13}{\mathrm{cm}}^{-2}$. (f) Band decomposed charge density at the point marked by the red square in (e).

Figure 4

(a) Band structure of ${\mathrm{Cu}}_7{\mathrm{B}}_{15}$. The band represented by yellow (blue) bubbles corresponds to the sum of contribution from B-$p_x$, B-$p_y$, Cu-$d_{xy}$, and Cu-$d_{x^2-y^2}$ (B-$p_z$, Cu-$d_{xz}$, and Cu-$d_{yz}$). The symbols “$+$” and “$-$” indicate the parity of the $z$-mirror symmetry operator for the bands $\alpha$ and $\beta$. (b) DOS of ${\mathrm{Cu}}_7{\mathrm{B}}_{15}$. (c) The 3D band structure. (d) The band gap of ${\mathrm{Cu}}_7{\mathrm{B}}_{15}$ in 2D reciprocal space around the $\mathrm{\Gamma }$ point. (e) Band structure of ${\mathrm{Cu}}_7{\mathrm{B}}_{15}$ with SOC. (f) Band structure of the slightly corrugated ${\mathrm{Cu}}_7{\mathrm{B}}_{15}$ without SOC.